The function $f(n)$ is defined as the product of all integers from 1 to $n,$ inclusive, and the function $g(n)$ is

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Thu Jun 04, 2020 7:26 am

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The function $f(n)$ is defined as the product of all integers from 1 to $n,$ inclusive, and the function $g(n)$ is defined as the product of all odd integers from 1 to $n,$ inclusive, where $n$ is a positive integer. If $p$ is a prime factor of $\dfrac{f(150)}{g(150)}+1,$ then which of the following must be true?

A. $p < 10$
B. $10 < p < 25$
C. $25 < p < 50$
D. $50 < p < 75$
E. $p > 75$

[spoiler]OA=E[/spoiler]

Source: e-GMAT

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Re: The function $f(n)$ is defined as the product of all integers from 1 to $n,$ inclusive, and the function \(g(n)\

by Scott@TargetTestPrep » Tue Jun 09, 2020 3:57 am

VJesus12 wrote: ↑
Thu Jun 04, 2020 7:26 am
The function $f(n)$ is defined as the product of all integers from 1 to $n,$ inclusive, and the function $g(n)$ is defined as the product of all odd integers from 1 to $n,$ inclusive, where $n$ is a positive integer. If $p$ is a prime factor of $\dfrac{f(150)}{g(150)}+1,$ then which of the following must be true?

A. $p < 10$
B. $10 < p < 25$
C. $25 < p < 50$
D. $50 < p < 75$
E. $p > 75$

[spoiler]OA=E[/spoiler]

Solution:

The quotient of the product of the first 150 positive integers (i.e., f(150)) and the product of the first 75 positive odd integers (i.e., g(150)) is the product of the first 75 positive even integers, i.e., f(150)/g(150) = 2 x 4 x 6 … x 148 x 150. The largest prime factor in this product is 73 (from the factor 146). However, when you add 1 to this product, that sum will have a prime factor greater than 73 since no two consecutive integers (notice that f(150)/g(150) and f(150)/g(150) + 1 are consecutive integers) share the same prime factors. Since a prime greater than 73 is also greater than 75, E is the correct answer.

Answer: E

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